Grade 12 · Math · Calculus

12th grade The Fundamental Theorem of Calculus worksheet

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The Fundamental Theorem of Calculus

Grade 12 Math Worksheet - Challenge Me

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Circle the best answer, or write your answer on the line.

Esme defines g(x) = ∫ from x to 9 of (t² + 1) dt, with x at the bottom. What is g′(3)?
Jun defines g(x) = ∫ from x to 9 of (t² + 4) dt, with x at the bottom. What is g′(2)?
Kira defines g(x) = ∫ from x to 9 of (t² + 2) dt, with x at the bottom. What is g′(6)?
Luz defines g(x) = ∫ from x to 9 of (t² + 6) dt, with x at the bottom. What is g′(1)?
Milo's rain barrel holds 20 liters. Water flows in at r(t) = 2t + 3 liters per minute. How much is in it after 4 minutes?
A pool slide's tank holds 50 gallons, and Noah fills it at r(t) = 4t + 1 gallons per minute. How much does it hold after 3 minutes?
Zee's sand pile has 100 kg, and sand is added at r(t) = 2t + 5 kg per hour. How much is there after 6 hours?
8.
If F′ = f, what does ∫ from a to b of f(x) dx equal, for Tala's notes?
- f(b) − f(a)
- F(b) − F(a)
- F(a) − F(b)
9.
Bo knows F(x) = x⁴/4 has F′(x) = x³. What is ∫ from 1 to 2 of x³ dx?
- f(2) − f(1) = 7
- F(2) − F(1) = 15/4
- F(1) − F(2) = −15/4
10.
What is the derivative of ∫ from 3 to x of sin t dt, for Ari's notes?
- sin x
- sin x − sin 3
- cos x

The Fundamental Theorem of Calculus, both parts. Every version has 10 questions, and the answer key prints on the second page.

I can find g′(x) for g(x) = ∫ from a to x of f(t) dt, including x² and x at the bottom.
I can evaluate a definite integral as F(b) − F(a).
I can say where an accumulation is largest or smallest from the sign of f.
I can find an amount from its start and the integral of its rate.

Standard: none (beyond CCSS)

accumulation function: g(x) = ∫ from a to x of f(t) dt: the area so far.

antiderivative: A function F with F′ = f.

moving end: The end of the integral that is x, not a number.

Subtracted f(b) − f(a) using the function itself. Part 2 needs an antiderivative F with F′ = f.

Added F(b) and F(a). The theorem takes the top minus the bottom.

Took the derivative of f. The derivative of the accumulation is f itself.

What is the most common mistake with the fundamental theorem of calculus?

Subtracted f(b) − f(a) using the function itself. Part 2 needs an antiderivative F with F′ = f.

Are these 12th grade math worksheets free to print?

Yes. All 6 printable versions at each level are free to print, with the answer key, and no account is needed.

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